Module 6 : Lecture 1 DIMENSIONAL ANALYSIS (Part I)

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1 Overview Modue 6 : Lecture DIMENSIONAL ANALYSIS (Part I) Many practica fow probes of different nature can be soved by using equations and anaytica procedures, as discussed in the previous odues. However, soutions of soe rea fow probes depend heaviy on experienta data and the refineents in the anaysis are ade, based on the easureents. Soeties, the experienta work in the aboratory is not ony tie-consuing, but aso expensive. So, the diensiona anaysis is an iportant too that heps in correating anaytica resuts with experienta data for such unknown fow probes. Aso, soe diensioness paraeters and scaing aws can be fraed in order to predict the prototype behavior fro the easureents on the ode. The iportant ters used in this odue ay be defined as beow; Diensiona Anaysis: The systeatic procedure of identifying the variabes in a physica phenoena and correating the to for a set of diensioness group is known as diensiona anaysis. Diensiona Hoogeneity: If an equation truy expresses a proper reationship aong variabes in a physica process, then it wi be diensionay hoogeneous. The equations are correct for any syste of units and consequenty each group of ters in the equation ust have the sae diensiona representation. This is aso known as the aw of diensiona hoogeneity. Diensiona variabes: These are the quantities, which actuay vary during a given case and can be potted against each other. Diensiona constants: These are noray hed constant during a given run. But, they ay vary fro case to case. Pure constants: They have no diensions, but, whie perforing the atheatica anipuation, they can arise. Joint initiative of IITs and IISc Funded by MHRD Page of 5

2 Let us expain these ters fro the foowing exapes: - Dispaceent of a free faing body is given as, S S t gt = + +, where, is the initia veocity, g is the acceeration due to gravity, t is the tie, S and S are the fina and initia distances, respectivey. Each ter in this equation has the diension of ength [ L] and hence it is diensionay hoogeneous. Here, S and t are the diensiona variabes, g, Sand are the diensiona constants and arises due to atheatica anipuation and is the pure constant. - Bernoui s equation for incopressibe fow is written as, p gz C ρ + + =. Here, p is the pressure, is the veocity, z is the distance, ρ is the density and g is the acceeration due to gravity. In this case, the diensiona variabes are p, and z, the diensiona constants are g, ρ and C and is the pure constant. Each ter in this equation incuding the constant has diension of diensionay hoogeneous. LT and hence it is Buckingha pi Theore The diensiona anaysis for the experienta data of unknown fow probes eads to soe non-diensiona paraeters. These diensioness products are frequenty referred as pi ters. Based on the concept of diensiona hoogeneity, these diensioness paraeters ay be grouped and expressed in functiona fors. This idea was expored by the faous scientist Edgar Buckingha (867-94) and the theore is naed accordingy. Buckingha pi theore, states that if an equation invoving k variabes is diensionay hoogeneous, then it can be reduced to a reationship aong ( k r) independent diensioness products, where r is the iniu nuber of reference diensions required to describe the variabe. For a physica syste, invoving k variabes, the functiona reation of variabes can be written atheaticay as, ( ) y = f x, x..., xk (6..) Joint initiative of IITs and IISc Funded by MHRD Page of 5

3 In Eq. (6..), it shoud be ensured that the diensions of the variabes on the eft side of the equation are equa to the diensions of any ter on the right side of equation. Now, it is possibe to rearrange the above equation into a set of diensioness products (pi ters), so that,..., k r Here, ϕ ( ) ( ) = ϕ, 3..., k r (6..) 3 is a function of through k r. The required nuber of pi ters is ess than the nuber of origina reference variabes by r. These reference diensions are usuay the basic diensions M, L and and Tie). T (Mass, Length Deterination of pi Ters Severa ethods can be used to for diensioness products or pi ters that arise in diensiona anaysis. But, there is a systeatic procedure caed ethod of repeating variabes that aows in deciding the diensioness and independent pi ters. For a given probe, foowing distinct steps are foowed. Step I: List out a the variabes that are invoved in the probe. The variabe is any quantity incuding diensiona and non-diensiona constants in a physica situation under investigation. Typicay, these variabes are those that are necessary to describe the geoetry of the syste (diaeter, ength etc.), to define fuid properties (density, viscosity etc.) and to indicate the externa effects infuencing the syste (force, pressure etc.). A the variabes ust be independent in nature so as to iniize the nuber of variabes required to describe the copete syste. Step II: Express each variabe in ters of basic diensions. Typicay, for fuid echanics probes, the basic diensions wi be either M, L and T or F, L and T. Diensionay, these two sets are reated through Newton s second aw ( F a. ) that F = MLT e.g. ρ 3 = ML or diensions shoud not be ixed. ρ = so 4 =. It shoud be noted that these basic FL T Step III: Decide the required nuber of pi ters. It can be deterined by using Buckingha pi theore which indicates that the nuber of pi ters is equa to ( k r), where k is the nuber of variabes in the probe (deterined fro Step I) and r is the nuber of reference diensions required to describe these variabes (deterined fro Step II). Joint initiative of IITs and IISc Funded by MHRD Page 3 of 5

4 Step I: Aongst the origina ist of variabes, seect those variabes that can be cobined to for pi ters. These are caed as repeating variabes. The required nuber of repeating variabes is equa to the nuber of reference diensions. Each repeating variabe ust be diensionay independent of the others, i.e. they cannot be cobined theseves to for any diensioness product. Since there is a possibiity of repeating variabes to appear in ore than one pi ter, so dependent variabes shoud not be chosen as one of the repeating variabe. Step : Essentiay, the pi ters are fored by utipying one of the non-repeating variabes by the product of the repeating variabes each raised to an exponent that wi a b c ake the cobination diensioness. It usuay takes the for of xi x x x 3 where the exponents a, b and care deterined so that the cobination is diensioness. Step I: Repeat the Step for each of the reaining non-repeating variabes. The resuting set of pi ters wi correspond to the required nuber obtained fro Step III. Step II: After obtaining the required nuber of pi ters, ake sure that a the pi ters are diensioness. It can be checked by sipy substituting the basic diension ( M, L and ) T of the variabes into the pi ters. Step III: Typicay, the fina for of reationship aong the pi ters can be written in the for of Eq. (6..) where, woud contain the dependent variabe in the nuerator. The actua functiona reationship aong pi ters is deterined fro experient. Joint initiative of IITs and IISc Funded by MHRD Page 4 of 5

5 Iustration of Pi Theore Let us consider the foowing exape to iustrate the procedure of deterining the various steps in the pi theore. Exape (Pressure drop in a pipe fow) Consider a steady fow of an incopressibe Newtonian fuid through a ong, sooth waed, horizonta circuar pipe. It is required to easure the pressure drop per unit ength of the pipe and find the nuber of non-diensiona paraeters invoved in the probe. Aso, it is desired to know the functiona reation aong these diensioness paraeters. Step I: Let us express a the pertinent variabes invoved in the experientation of pressure drop per unit ength ( p ) of the pipe, in the foowing for; (, ρµ,, ) p = f D (6..3) where, D is the pipe diaeter, ρ is the fuid density, µ is the viscosity of the fuid and is the ean veocity at which the fuid is fowing through the pipe. Step II: Next step is to express a the variabes in ters of basic diensions i.e. M, L and T. It then foows that 3 p = ML T ; D = L; ρ = ML ; µ = ML T ; = LT (6..4) Step III: Appy Buckingha theore to decide the nuber of pi ters required. There are five variabes (incuding the dependent variabe p ) and three reference diensions. Since, k = 5 and r = 3, ony two pi ters are required for this probe. Step I: The repeating variabes to for pi ters, need to be seected fro the ist D, ρ, µ and. It is to be noted that the dependent variabe shoud not be used as one of the repeating variabe. Since, there are three reference diensions invoved, so we need to seect three repeating variabe. These repeating variabes shoud be diensionay independent, i.e. diensioness product cannot be fored fro this set. In this case, D, ρ and ay be chosen as the repeating variabes. Step : Now, first pi ter is fored between the dependent variabe and the repeating variabes. It is written as, a b c = pdρ (6..5) Since, this cobination need to be diensioness, it foows that b a ( )( ) ( ) ( ) c 3 ML T L LT ML = M L T (6..6) Joint initiative of IITs and IISc Funded by MHRD Page 5 of 5

6 The exponents a, b and c ust be deterined by equating the exponents for each of the ters M, L and T i.e. For M : + c= For L: + a+ b 3c= For T : b= The soution of this agebraic equations gives a = ; b= ; c=. Therefore, (6..7) pd = (6..8) ρ The process is repeated for reaining non-repeating variabes with other additiona variabe ( µ ) so that, d e f = µ. D.. ρ (6..9) Since, this cobination need to be diensioness, it foows that Equating the exponents, e d ( )( ) ( ) ( ) f 3 ML T L LT ML = M L T (6..) For M : + f = For L: + d + e 3 f = For T : e= The soution of this agebraic equation gives d = ; e= ; f =. Therefore, (6..) µ = (6..) ρd Step I: Now, the correct nubers of pi ters are fored as deterined in Step III. In order to ake sure about the diensionaity of pi ters, they are written as, ( ML T )( L) pd = = = M LT ρ ML LT 3 ( )( ) ( ML T )( L) ( )( )( ) µ = = = M LT 3 ρd ML LT L (6..3) Step II: Finay, the resut of diensiona anaysis is expressed aong the pi ters as, D p µ = φ φ = ρ ρd Re It ay be noted here that Re is the Reynods nuber. (6..4) Joint initiative of IITs and IISc Funded by MHRD Page 6 of 5

7 Rearks - If the difference in the nuber of variabes for a given probe and nuber of reference diensions is equa to unity, then ony one Pi ter is required to describe the phenoena. Here, the functiona reationship for the one Pi ter is a constant quantity and it is deterined fro the experient. = Constant (6..5) - The probes invoving two Pi ters can be described such that ( ) = φ (6..6) Here, the functiona reationship aong the variabes can then be deterined by varying and easuring the corresponding vaues of. Joint initiative of IITs and IISc Funded by MHRD Page 7 of 5

8 Modue 6 : Lecture DIMENSIONAL ANALYSIS (Part II) Non Diensiona nubers in Fuid Dynaics Forces encountered in fowing fuids incude those due to inertia, viscosity, pressure, gravity, surface tension and copressibiity. These forces can be written as foows; d Inertia force: a. = ρ ρ L dt du iscous force: τ A= µ A µ L dy Pressure force: ( p) A ( p) L (6..) 3 Gravity force: g gρ L Surface tension force: σ L Copressibiity force: Ev A Ev L The notations used in Eq. (6..) are given in subsequent paragraph of this section. It ay be noted that the ratio of any two forces wi be diensioness. Since, inertia forces are very iportant in fuid echanics probes, the ratio of the inertia force to each of the other forces isted above eads to fundaenta diensioness groups. Soe of the are defined as given beow; Reynods nuber ( Re ) : It is defined as the ratio of inertia force to viscous force. Matheaticay, Re ρl L = = (6..) µ ν where is the veocity of the fow, L is the characteristics ength, ρ, µ and ν are the density, dynaic viscosity and kineatic viscosity of the fuid respectivey. If Re is very sa, there is an indication that the viscous forces are doinant copared to inertia forces. Such types of fows are coony referred to as creeping/viscous fows. Conversey, for arge Re, viscous forces are sa copared to inertia effects and such fow probes are characterized as inviscid anaysis. This nuber is aso used to study the transition between the ainar and turbuent fow regies. Joint initiative of IITs and IISc Funded by MHRD Page 8 of 5

9 Euer nuber ( E u ) : In ost of the aerodynaic ode testing, the pressure data are usuay expressed atheaticay as, where E u p = (6..3) ρ p is the difference in oca pressure and free strea pressure, is the veocity of the fow, ρ is the density of the fuid. The denoinator in Eq. (6..3) is caed dynaic pressure. E u is the ratio of pressure force to inertia force and any a ties the pressure coefficient ( cp ) is a aso coon nae which is defined by sae anner. In the study of cavitations phenoena, siiar expressions are used where, p is the difference in iquid strea pressure and iquid-vapour pressure. This diensiona paraeter is then caed as cavitation nuber. Froude nuber ( F r ) : It is interpreted as the ratio of inertia force to gravity force. Matheaticay, it is written as, Fr = (6..4) gl. where is the veocity of the fow, L is the characteristics ength descriptive of the fow fied and g is the acceeration due to gravity. This nuber is very uch significant for fows with free surface effects such as in case of open-channe fow. In such types of fows, the characteristics ength is the depth of water. F r ess than unity indicates sub-critica fow and vaues greater than unity indicate super-critica fow. It is aso used to study the fow of water around ships with resuting wave otion. Weber nuber ( W e ) : It is defined as the ratio of the inertia force to surface tension force. Matheaticay, W e ρ L = (6..5) σ where is the veocity of the fow, L is the characteristics ength descriptive of the fow fied, ρ is the density of the fuid and σ is the surface tension force. This nuber is taken as an index of dropet foration and fow of thin fi iquids in which there is an interface between two fuids. The inertia force is doinant copared to surface tension force when, We (e.g. fow of water in a river). Joint initiative of IITs and IISc Funded by MHRD Page 9 of 5

10 Mach nuber ( M ) : It is the key paraeter that characterizes the copressibiity effects in a fuid fow and is defined as the ratio of inertia force to copressibiity force. Matheaticay, M = = = (6..6) c dp E v d ρ ρ where is the veocity of the fow, c is the oca sonic speed, ρ is the density of the fuid and E v is the buk oduus. Soeties, the square of the Mach nuber is caed Cauchy nuber ( C a ) i.e. C ρ a = M = (6..7) Ev Both the nubers are predoinanty used in probes in which fuid copressibiity is iportant. When, M a is reativey sa (say, ess than.3), the inertia forces induced by fuid otion are sufficienty sa to cause significant change in fuid density. So, the copressibiity of the fuid can be negected. However, this nuber is ost coony used paraeter in copressibe fuid fow probes, particuary in the fied of gas dynaics and aerodynaics. Strouha nuber ( S t ) : It is a diensioness paraeter that is ikey to be iportant in unsteady, osciating fow probes in which the frequency of osciation is ω and is defined as, ωl St = (6..8) where is the veocity of the fow and L is the characteristics ength descriptive of the fow fied. This nuber is the easure of the ratio of the inertia forces due to unsteadiness of the fow (oca acceeration) to inertia forces due to changes in veocity fro point to point in the fow fied (convective acceeration). This type of unsteady fow deveops when a fuid fows past a soid body paced in the oving strea. Joint initiative of IITs and IISc Funded by MHRD Page of 5

11 In addition, there are few other diensioness nubers that are of iportance in fuid echanics. They are isted beow; Paraeter Matheatica expression Quaitative definition Iportance Prandt nuber P r µ c = k p Dissipation Conduction Heat convection Eckert nuber E c = c T p Kinetic energy Enthapy Dissipation Specific heat ratio c γ = c p v Enthapy Interna energy Copressibe fow Roughness ratio ε L Wa roughness Body ength Turbuent rough was Grashof nuber G r β = 3 ( T) glρ µ Buoyancy iscosity Natura onvection Teperature ratio Pressure coefficient Aerodynaics Lift coefficient Drag coefficient dynaics T w T C C C p L D = = = Aero dynaics p p ( ) ρ L ( ) Aρ D ( ) Aρ Wa teperature Strea teperature Static pressure Dynaic pressure Lift force Dynaic force Drag force Dynaic force Heat transfer Hydrodynaics, Hydrodynaics,Aero Hydrodynaics, Joint initiative of IITs and IISc Funded by MHRD Page of 5

12 Modeing and Siiitude A ode is a representation of a physica syste which is used to predict the behavior of the syste in soe desired respect. The physica syste for which the predictions are to be ade is caed prototype. Usuay, a ode is saer than the prototype so that aboratory experients/studies can be conducted. It is ess expensive to construct and operate. However, in certain situations, odes are arger than the prototype e.g. study of the otion of bood ces whose sizes are of the order of icroeters. Siiitude is the indication of a known reationship between a ode and prototype. In other words, the ode tests ust yied data that can be scaed to obtain the siiar paraeters for the prototype. Theory of odes: The diensiona anaysis of a given probe can be described in ters of a set of pi ters and these non-diensiona paraeters can be expressed in functiona fors; ( ) = φ,,... n (6..9) 3 Since this equation appies to any syste, governed by sae variabes and if the behavior of a particuar prototype is described by Eq. (6..9), then a siiar reationship can be written for a ode. (,,... ) = φ (6..) 3 n The for of the function reains the sae as ong as the sae phenoenon is invoved in both the prototype and the ode. Therefore, if the ode is designed and operated under foowing conditions, Then it foows that = ; =... and = (6..) 3 3 n n = (6..) Eq. (6..) is the desired prediction equation and indicates that the easured vaue of obtained with the ode wi be equa to the corresponding for the prototype as ong as the other pi ters are equa. These are caed ode design conditions / siiarity requireents / odeing aws. Joint initiative of IITs and IISc Funded by MHRD Page of 5

13 Fow Siiarity In order to achieve siiarity between ode and prototype behavior, a the corresponding pi ters ust be equated to satisfy the foowing conditions. Geoetric siiarity: A ode and prototype are geoetric siiar if and ony if a body diensions in a three coordinates have the sae inear-scae ratio. In order to have geoetric siiarity between the ode and prototype, the ode and the prototype shoud be of the sae shape, a the inear diensions of the ode can be reated to corresponding diensions of the prototype by a constant scae factor. Usuay, one or ore of these pi ters wi invove ratios of iportant engths, which are purey geoetrica in nature. Kineatic siiarity: The otions of two systes are kineaticay siiar if hoogeneous partices ie at sae points at sae ties. In a specific sense, the veocities at corresponding points are in the sae direction (i.e. sae streaine patterns) and are reated in agnitude by a constant scae factor. Dynaic siiarity: When two fows have force distributions such that identica types of forces are parae and are reated in agnitude by a constant scae factor at a corresponding points, then the fows are dynaic siiar. For a ode and prototype, the dynaic siiarity exists, when both of the have sae ength-scae ratio, tiescae ratio and force-scae (or ass-scae ratio). In order to have copete siiarity between the ode and prototype, a the siiarity fow conditions ust be aintained. This wi autoaticay foow if a the iportant variabes are incuded in the diensiona anaysis and if a the siiarity requireents based on the resuting pi ters are satisfied. For exape, in copressibe fows, the ode and prototype shoud have sae Reynods nuber, Mach nuber and specific heat ratio etc. If the fow is incopressibe (without free surface), then sae Reynods nubers for ode and prototype can satisfy the copete siiarity. Joint initiative of IITs and IISc Funded by MHRD Page 3 of 5

14 Mode scaes In a given probe, if there are two ength variabes and, the resuting requireent based on the pi ters obtained fro these variabes is, = = λ (6..3) This ratio is defined as the ength scae. For true odes, there wi be ony one ength scae and a engths are fixed in accordance with this scae. There are other ode scaes such as veocity scae = λ v, density scae ρ = λρ ρ, viscosity scae µ = λµ µ etc. Each of these scaes needs to be defined for a given probe. Distorted odes In order to achieve the copete dynaic siiarity between geoetricay siiar fows, it is necessary to reproduce the independent diensioness groups so that dependent paraeters can aso be dupicated (e.g. sae Reynods nuber between a ode and prototype is ensured for dynaicay siiar fows). In any ode studies, dynaic siiarity ay aso ead to incopete siiarity between the ode and the prototype. If one or ore of the siiarity requireents are not et, e.g. in Eq. 6..9, if, then it foows that Eq. 6.. wi not be satisfied i.e.. It is a case of distorted ode for which one or ore of the siiar requireents are not satisfied. For exape, in the study of free surface fows, ρ both Reynods nuber µ and Froude nuber are invoved. Then, g Froude nuber siiarity requires, = (6..4) g g If the ode and prototype are operated in the sae gravitationa fied, then the veocity scae becoes, = = λ (6..5) Joint initiative of IITs and IISc Funded by MHRD Page 4 of 5

15 Reynods nuber siiarity requires, Then, the veocity scae is, ρ.. ρ.. = (6..6) µ µ µ ρ =.. (6..7) µ ρ Since, the veocity scae ust be equa to the square root of the ength scae, it foows that ν ν 3 = = = ( µ ρ) ( µ ρ) ( λ ) 3 (6..8) Eq. (6..8) requires that both ode and prototype to have different kineatics viscosity scae. But practicay, it is aost ipossibe to find a suitabe fuid for the ode, in sa ength scae. In such cases, the systes are designed on the basis of Froude nuber with different Reynods nuber for the ode and prototype where Eq. (6..8) need not be satisfied. Such anaysis wi resut a distorted ode and there are no genera rues for handing distorted odes, rather each probe ust be considered on its own erits. Joint initiative of IITs and IISc Funded by MHRD Page 5 of 5